Sizing¶

Size a solar heating system - let the optimizer decide equipment sizes.

This notebook introduces:

InvestParameters: Define investment decisions with size bounds and costs
Investment costs: Fixed costs and size-dependent costs
Optimal sizing: Let the optimizer find the best equipment sizes
Trade-off analysis: Balance investment vs. operating costs

Setup¶

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import numpy as np
import pandas as pd
import plotly.express as px
import xarray as xr

import flixopt as fx

fx.CONFIG.notebook()
import numpy as np import pandas as pd import plotly.express as px import xarray as xr import flixopt as fx fx.CONFIG.notebook()

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flixopt.config.CONFIG

System Description¶

The swimming pool heating system:

Solar collectors: Convert solar radiation to heat (size to be optimized)
Gas boiler: Backup heating when solar is insufficient (existing, 200 kW)
Buffer tank: Store excess solar heat (size to be optimized)
Pool: Constant heat demand of 150 kW during operating hours

   ☀️ Solar ──► [Heat] ◄── Boiler ◄── [Gas]
                  │
                  ▼
              Buffer Tank
                  │
                  ▼
                Pool 🏊

Define Time Horizon and Profiles¶

We model one representative summer week:

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# One week in summer, hourly
timesteps = pd.date_range('2024-07-15', periods=168, freq='h')
hours = np.arange(168)
hour_of_day = hours % 24

# Solar radiation profile (kW/m² equivalent, simplified)
# Peak around noon, zero at night
solar_profile = np.maximum(0, np.sin((hour_of_day - 6) * np.pi / 12)) * 0.8
solar_profile = np.where((hour_of_day >= 6) & (hour_of_day <= 20), solar_profile, 0)

# Add some cloud variation
np.random.seed(42)
cloud_factor = np.random.uniform(0.6, 1.0, len(timesteps))
solar_profile = solar_profile * cloud_factor

# Pool operates 8am-10pm, constant demand when open
pool_demand = np.where((hour_of_day >= 8) & (hour_of_day <= 22), 150, 50)  # kW

print(f'Peak solar: {solar_profile.max():.2f} kW/kW_installed')
print(f'Pool demand: {pool_demand.max():.0f} kW (open), {pool_demand.min():.0f} kW (closed)')
# One week in summer, hourly timesteps = pd.date_range('2024-07-15', periods=168, freq='h') hours = np.arange(168) hour_of_day = hours % 24 # Solar radiation profile (kW/m² equivalent, simplified) # Peak around noon, zero at night solar_profile = np.maximum(0, np.sin((hour_of_day - 6) * np.pi / 12)) * 0.8 solar_profile = np.where((hour_of_day >= 6) & (hour_of_day <= 20), solar_profile, 0) # Add some cloud variation np.random.seed(42) cloud_factor = np.random.uniform(0.6, 1.0, len(timesteps)) solar_profile = solar_profile * cloud_factor # Pool operates 8am-10pm, constant demand when open pool_demand = np.where((hour_of_day >= 8) & (hour_of_day <= 22), 150, 50) # kW print(f'Peak solar: {solar_profile.max():.2f} kW/kW_installed') print(f'Pool demand: {pool_demand.max():.0f} kW (open), {pool_demand.min():.0f} kW (closed)')

Peak solar: 0.76 kW/kW_installed
Pool demand: 150 kW (open), 50 kW (closed)

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# Visualize profiles with plotly - using xarray and faceting
profiles = xr.Dataset(
    {
        'Solar Profile [kW/kW]': xr.DataArray(solar_profile, dims=['time'], coords={'time': timesteps}),
        'Pool Demand [kW]': xr.DataArray(pool_demand, dims=['time'], coords={'time': timesteps}),
    }
)

# Convert to long format for faceting
df = profiles.to_dataframe().reset_index().melt(id_vars='time', var_name='variable', value_name='value')
fig = px.line(df, x='time', y='value', facet_col='variable', height=300)
fig.update_yaxes(matches=None, showticklabels=True)
fig.for_each_annotation(lambda a: a.update(text=a.text.split('=')[-1]))
fig
# Visualize profiles with plotly - using xarray and faceting profiles = xr.Dataset( { 'Solar Profile [kW/kW]': xr.DataArray(solar_profile, dims=['time'], coords={'time': timesteps}), 'Pool Demand [kW]': xr.DataArray(pool_demand, dims=['time'], coords={'time': timesteps}), } ) # Convert to long format for faceting df = profiles.to_dataframe().reset_index().melt(id_vars='time', var_name='variable', value_name='value') fig = px.line(df, x='time', y='value', facet_col='variable', height=300) fig.update_yaxes(matches=None, showticklabels=True) fig.for_each_annotation(lambda a: a.update(text=a.text.split('=')[-1])) fig

Define Costs¶

Investment costs are annualized (€/year) to compare with operating costs:

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# Cost parameters
GAS_PRICE = 0.12  # €/kWh - high gas price makes solar attractive

# Solar collectors: 400 €/kW installed, 20-year lifetime → ~25 €/kW/year annualized
# (simplified, real calculation would include interest rate)
SOLAR_COST_PER_KW = 20  # €/kW/year

# Buffer tank: 50 €/kWh capacity, 30-year lifetime → ~2 €/kWh/year
TANK_COST_PER_KWH = 1.5  # €/kWh/year

# Scale factor: We model 1 week, but costs are annual
# So we scale investment costs to weekly equivalent
WEEKS_PER_YEAR = 52
SOLAR_COST_WEEKLY = SOLAR_COST_PER_KW / WEEKS_PER_YEAR
TANK_COST_WEEKLY = TANK_COST_PER_KWH / WEEKS_PER_YEAR

print(f'Solar cost: {SOLAR_COST_WEEKLY:.3f} €/kW/week')
print(f'Tank cost: {TANK_COST_WEEKLY:.4f} €/kWh/week')
# Cost parameters GAS_PRICE = 0.12 # €/kWh - high gas price makes solar attractive # Solar collectors: 400 €/kW installed, 20-year lifetime → ~25 €/kW/year annualized # (simplified, real calculation would include interest rate) SOLAR_COST_PER_KW = 20 # €/kW/year # Buffer tank: 50 €/kWh capacity, 30-year lifetime → ~2 €/kWh/year TANK_COST_PER_KWH = 1.5 # €/kWh/year # Scale factor: We model 1 week, but costs are annual # So we scale investment costs to weekly equivalent WEEKS_PER_YEAR = 52 SOLAR_COST_WEEKLY = SOLAR_COST_PER_KW / WEEKS_PER_YEAR TANK_COST_WEEKLY = TANK_COST_PER_KWH / WEEKS_PER_YEAR print(f'Solar cost: {SOLAR_COST_WEEKLY:.3f} €/kW/week') print(f'Tank cost: {TANK_COST_WEEKLY:.4f} €/kWh/week')

Solar cost: 0.385 €/kW/week
Tank cost: 0.0288 €/kWh/week

Build the System with Investment Options¶

Use InvestParameters to define which sizes should be optimized:

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flow_system = fx.FlowSystem(timesteps)
flow_system.add_carriers(
    fx.Carrier('gas', '#3498db', 'kW'),
    fx.Carrier('heat', '#e74c3c', 'kW'),
)
flow_system.add_elements(
    # === Buses ===
    fx.Bus('Heat', carrier='heat'),
    fx.Bus('Gas', carrier='gas'),
    # === Effects ===
    fx.Effect('costs', '€', 'Total Costs', is_standard=True, is_objective=True),
    # === Gas Supply ===
    fx.Source(
        'GasGrid',
        outputs=[fx.Flow('Gas', bus='Gas', size=500, effects_per_flow_hour=GAS_PRICE)],
    ),
    # === Gas Boiler (existing, fixed size) ===
    fx.linear_converters.Boiler(
        'GasBoiler',
        thermal_efficiency=0.92,
        thermal_flow=fx.Flow('Heat', bus='Heat', size=200),  # 200 kW existing
        fuel_flow=fx.Flow('Gas', bus='Gas'),
    ),
    # === Solar Collectors (size to be optimized) ===
    fx.Source(
        'SolarCollectors',
        outputs=[
            fx.Flow(
                'Heat',
                bus='Heat',
                # Investment optimization: find optimal size between 0-500 kW
                size=fx.InvestParameters(
                    minimum_size=0,
                    maximum_size=500,
                    effects_of_investment_per_size={'costs': SOLAR_COST_WEEKLY},
                ),
                # Solar output depends on radiation profile
                fixed_relative_profile=solar_profile,
            )
        ],
    ),
    # === Buffer Tank (size to be optimized) ===
    fx.Storage(
        'BufferTank',
        # Investment optimization: find optimal capacity between 0-2000 kWh
        capacity_in_flow_hours=fx.InvestParameters(
            minimum_size=0,
            maximum_size=2000,
            effects_of_investment_per_size={'costs': TANK_COST_WEEKLY},
        ),
        initial_charge_state=0,
        eta_charge=0.95,
        eta_discharge=0.95,
        relative_loss_per_hour=0.01,  # 1% loss per hour
        charging=fx.Flow('Charge', bus='Heat', size=200),
        discharging=fx.Flow('Discharge', bus='Heat', size=200),
    ),
    # === Pool Heat Demand ===
    fx.Sink(
        'Pool',
        inputs=[fx.Flow('Heat', bus='Heat', size=1, fixed_relative_profile=pool_demand)],
    ),
)
flow_system = fx.FlowSystem(timesteps) flow_system.add_carriers( fx.Carrier('gas', '#3498db', 'kW'), fx.Carrier('heat', '#e74c3c', 'kW'), ) flow_system.add_elements( # === Buses === fx.Bus('Heat', carrier='heat'), fx.Bus('Gas', carrier='gas'), # === Effects === fx.Effect('costs', '€', 'Total Costs', is_standard=True, is_objective=True), # === Gas Supply === fx.Source( 'GasGrid', outputs=[fx.Flow('Gas', bus='Gas', size=500, effects_per_flow_hour=GAS_PRICE)], ), # === Gas Boiler (existing, fixed size) === fx.linear_converters.Boiler( 'GasBoiler', thermal_efficiency=0.92, thermal_flow=fx.Flow('Heat', bus='Heat', size=200), # 200 kW existing fuel_flow=fx.Flow('Gas', bus='Gas'), ), # === Solar Collectors (size to be optimized) === fx.Source( 'SolarCollectors', outputs=[ fx.Flow( 'Heat', bus='Heat', # Investment optimization: find optimal size between 0-500 kW size=fx.InvestParameters( minimum_size=0, maximum_size=500, effects_of_investment_per_size={'costs': SOLAR_COST_WEEKLY}, ), # Solar output depends on radiation profile fixed_relative_profile=solar_profile, ) ], ), # === Buffer Tank (size to be optimized) === fx.Storage( 'BufferTank', # Investment optimization: find optimal capacity between 0-2000 kWh capacity_in_flow_hours=fx.InvestParameters( minimum_size=0, maximum_size=2000, effects_of_investment_per_size={'costs': TANK_COST_WEEKLY}, ), initial_charge_state=0, eta_charge=0.95, eta_discharge=0.95, relative_loss_per_hour=0.01, # 1% loss per hour charging=fx.Flow('Charge', bus='Heat', size=200), discharging=fx.Flow('Discharge', bus='Heat', size=200), ), # === Pool Heat Demand === fx.Sink( 'Pool', inputs=[fx.Flow('Heat', bus='Heat', size=1, fixed_relative_profile=pool_demand)], ), )

Run Optimization¶

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flow_system.optimize(fx.solvers.HighsSolver(mip_gap=0.01));
flow_system.optimize(fx.solvers.HighsSolver(mip_gap=0.01));

Running HiGHS 1.12.0 (git hash: 755a8e0): Copyright (c) 2025 HiGHS under MIT licence terms
MIP linopy-problem-__3fw0hg has 2712 rows; 2374 cols; 7842 nonzeros; 338 integer variables (338 binary)
Coefficient ranges:
  Matrix  [1e-05, 2e+03]
  Cost    [1e+00, 1e+00]
  Bound   [3e-14, 2e+03]
  RHS     [1e+00, 1e+00]
WARNING: Problem has some excessively small column bounds
Presolving model
1348 rows, 1012 cols, 3276 nonzeros  0s
1013 rows, 677 cols, 3673 nonzeros  0s
1012 rows, 676 cols, 3672 nonzeros  0s
Presolve reductions: rows 1012(-1700); columns 676(-1698); nonzeros 3672(-4170) 

Solving MIP model with:
   1012 rows
   676 cols (338 binary, 0 integer, 0 implied int., 338 continuous, 0 domain fixed)
   3672 nonzeros

Src: B => Branching; C => Central rounding; F => Feasibility pump; H => Heuristic;
     I => Shifting; J => Feasibility jump; L => Sub-MIP; P => Empty MIP; R => Randomized rounding;
     S => Solve LP; T => Evaluate node; U => Unbounded; X => User solution; Y => HiGHS solution;
     Z => ZI Round; l => Trivial lower; p => Trivial point; u => Trivial upper; z => Trivial zero

        Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work      
Src  Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

 J       0       0         0   0.00%   -inf            1690.076848        Large        0      0      0         0     0.0s
         0       0         0   0.00%   782.9837399     1690.076848       53.67%        0      0      0       364     0.0s
 C       0       0         0   0.00%   782.9837399     1488.93016        47.41%     1537     54      0       418     0.1s

 L       0       0         0   0.00%   782.9837399     782.9837399        0.00%     2396     82      0       454     0.3s
         1       0         1 100.00%   782.9837399     782.9837399        0.00%     2396     82      0       794     0.3s

Solving report
  Model             linopy-problem-__3fw0hg
  Status            Optimal
  Primal bound      782.983739927
  Dual bound        782.983739927
  Gap               0% (tolerance: 1%)
  P-D integral      0.151347858271
  Solution status   feasible
                    782.983739927 (objective)
                    0 (bound viol.)
                    3.46389583683e-14 (int. viol.)
                    0 (row viol.)
  Timing            0.32
  Max sub-MIP depth 1
  Nodes             1
  Repair LPs        0
  LP iterations     794
                    0 (strong br.)
                    90 (separation)
                    340 (heuristics)

Analyze Investment Decisions¶

Optimal Sizes¶

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solar_size = flow_system.statistics.sizes['SolarCollectors(Heat)'].item()
tank_size = flow_system.statistics.sizes['BufferTank'].item()

print('=== Optimal Investment Decisions ===')
print(f'Solar collectors: {solar_size:.1f} kW')
print(f'Buffer tank: {tank_size:.1f} kWh')
print(f'Tank-to-solar ratio: {tank_size / solar_size:.1f} kWh/kW' if solar_size > 0 else 'N/A')
solar_size = flow_system.statistics.sizes['SolarCollectors(Heat)'].item() tank_size = flow_system.statistics.sizes['BufferTank'].item() print('=== Optimal Investment Decisions ===') print(f'Solar collectors: {solar_size:.1f} kW') print(f'Buffer tank: {tank_size:.1f} kWh') print(f'Tank-to-solar ratio: {tank_size / solar_size:.1f} kWh/kW' if solar_size > 0 else 'N/A')

=== Optimal Investment Decisions ===
Solar collectors: 458.5 kW
Buffer tank: 747.7 kWh
Tank-to-solar ratio: 1.6 kWh/kW

Visualize Sizes¶

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flow_system.statistics.plot.sizes()
flow_system.statistics.plot.sizes()

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Cost Breakdown¶

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total_costs = flow_system.solution['costs'].item()

# Calculate cost components
solar_invest = solar_size * SOLAR_COST_WEEKLY
tank_invest = tank_size * TANK_COST_WEEKLY
gas_costs = total_costs - solar_invest - tank_invest

print('=== Weekly Cost Breakdown ===')
print(f'Solar investment: {solar_invest:.2f} € ({solar_invest / total_costs * 100:.1f}%)')
print(f'Tank investment:  {tank_invest:.2f} € ({tank_invest / total_costs * 100:.1f}%)')
print(f'Gas operating:    {gas_costs:.2f} € ({gas_costs / total_costs * 100:.1f}%)')
print('─────────────────────────────')
print(f'Total:            {total_costs:.2f} €')
total_costs = flow_system.solution['costs'].item() # Calculate cost components solar_invest = solar_size * SOLAR_COST_WEEKLY tank_invest = tank_size * TANK_COST_WEEKLY gas_costs = total_costs - solar_invest - tank_invest print('=== Weekly Cost Breakdown ===') print(f'Solar investment: {solar_invest:.2f} € ({solar_invest / total_costs * 100:.1f}%)') print(f'Tank investment: {tank_invest:.2f} € ({tank_invest / total_costs * 100:.1f}%)') print(f'Gas operating: {gas_costs:.2f} € ({gas_costs / total_costs * 100:.1f}%)') print('─────────────────────────────') print(f'Total: {total_costs:.2f} €')

=== Weekly Cost Breakdown ===
Solar investment: 176.33 € (22.5%)
Tank investment:  21.57 € (2.8%)
Gas operating:    585.09 € (74.7%)
─────────────────────────────
Total:            782.98 €

System Operation¶

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flow_system.statistics.plot.balance('Heat')
flow_system.statistics.plot.balance('Heat')

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flow_system.statistics.plot.heatmap('SolarCollectors(Heat)')
flow_system.statistics.plot.heatmap('SolarCollectors(Heat)')

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flow_system.statistics.plot.balance('BufferTank')
flow_system.statistics.plot.balance('BufferTank')

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Compare: What if No Solar?¶

Let's see how much the solar system saves:

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# Gas-only scenario
total_demand = pool_demand.sum()
gas_only_cost = total_demand / 0.92 * GAS_PRICE  # All heat from gas boiler

savings = gas_only_cost - total_costs
savings_pct = savings / gas_only_cost * 100

print('=== Comparison with Gas-Only ===')
print(f'Gas-only cost: {gas_only_cost:.2f} €/week')
print(f'With solar:    {total_costs:.2f} €/week')
print(f'Savings:       {savings:.2f} €/week ({savings_pct:.1f}%)')
print(f'Annual savings: {savings * 52:.0f} €/year')
# Gas-only scenario total_demand = pool_demand.sum() gas_only_cost = total_demand / 0.92 * GAS_PRICE # All heat from gas boiler savings = gas_only_cost - total_costs savings_pct = savings / gas_only_cost * 100 print('=== Comparison with Gas-Only ===') print(f'Gas-only cost: {gas_only_cost:.2f} €/week') print(f'With solar: {total_costs:.2f} €/week') print(f'Savings: {savings:.2f} €/week ({savings_pct:.1f}%)') print(f'Annual savings: {savings * 52:.0f} €/year')

=== Comparison with Gas-Only ===
Gas-only cost: 2465.22 €/week
With solar:    782.98 €/week
Savings:       1682.23 €/week (68.2%)
Annual savings: 87476 €/year

Energy Flow Sankey¶

A Sankey diagram visualizes the total energy flows through the system:

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flow_system.statistics.plot.sankey.flows()
flow_system.statistics.plot.sankey.flows()

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Key Concepts¶

InvestParameters Options¶

fx.InvestParameters(
    minimum_size=0,           # Lower bound (can be 0 for optional)
    maximum_size=500,         # Upper bound
    fixed_size=100,           # Or: fixed size (binary decision)
    mandatory=True,           # Force investment to happen
    effects_of_investment={'costs': 1000},      # Fixed cost if invested
    effects_of_investment_per_size={'costs': 25},  # Cost per unit size
)

Where to Use InvestParameters¶

Flow.size: Optimize converter/source/sink capacity
Storage.capacity_in_flow_hours: Optimize storage capacity

Summary¶

You learned how to:

Define investment decisions with InvestParameters
Set size bounds (minimum/maximum)
Add investment costs (per-size and fixed)
Access optimal sizes via statistics.sizes
Visualize sizes with statistics.plot.sizes()

Next Steps¶

04-operational-constraints: Add startup costs and minimum run times
05-multi-carrier-system: Model combined heat and power